A fullerene is a spherical compound consisting of only carbon atoms. The term includes higher order fullerenes consisting of 60 carbon atoms (C60) and higher even numbers of carbon atoms. These include 12 five-member rings and 20 or more five-member rings, six-member rings or seven-member rings. It has been learned in recent years that the most representative C60 is a molecule extremely high in reactivity due to its special electron system. This reactivity is utilized for various types of chemical modification. Blending a spherical carbon molecule fullerene into a rubber component, as a filler, is described in, for example, Japanese Patent Publication (A) No. 10-168238. Further, polymerizing isoprene by living anion polymerization and modifying its terminals by a fullerene is described in, for example, Macromolecules 1997, 30, 2546-2555. However, it has not been known that the conjugated diene-based polymer synthesized by the method described in Macromolecules is useful as a rubber composition.